Relation between Fresnel IOR and Layerweight blend?
By comparing visual results from both LayerWeight's Fresnel and Fresnel Node on a test sphere I concluded that these settings give similar results:



Blend:  0.00  0.10  0.50  0.70  0.80  0.90  0.95  0.98  1.00
  IOR:  1.00  1.10  1.70  3.30  5.10  10.0  20.0  50.0  1000


The Y scale is logarithmic

Confirmed by Brecht (thanks) the conversion formula is:


IOR = 1/(1 - Blend)
Blend = 1 - 1/IOR




Layer Weight node:


Layer Weight Facing output mixes the two shaders based of angle of incidence.
The Fresnel output mixes them based on fresnel formula (Dielectric fresnel weight), which is a function of angle of incidence. The exact math is here: Wiki
The Blend slider adds additional blending between the two shaders: at 0 it will be green and at 1 it will be red.




On the other hand the Fresnel node allows for proper IOR input:



I use Fresnel node when dealing with glass-like materials, where the light goes through - where it bends at the interface. I can set proper IOR for the interface this way.

I use Layer Weight node when dealing with reflectivity, it allows for better control over the transition on the surface (how smooth it is) and over how the materials are mixed (with that blend slider). This suits artistic approach better when you are eye-balling stuff.